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Home , Quantum metrology

Quantum metrology: An information-theoretic perspective

in three two lectures

Carlton M. Caves Center for and Control, University of New Mexico

http://info.phys.unm.edu/~caves

Center for Quantum Information and Control Quantum metrology: An information-theoretic perspective

Lecture 1 I. Introduction. What’s the problem? II. Squeezed states and optical interferometry III. Ramsey interferometry, cat states, and squeezing

Carlton M. Caves Center for Quantum Information and Control, University of New Mexico

http://info.phys.unm.edu/~caves

Center for Quantum Information and Control I. Introduction. What’s the problem?

View from Cape Hauy Tasman Peninsula Tasmania Quantum information science

A new way of thinking Computer science Computational complexity depends on physical law.

New physics Old physics as liberator. Quantum mechanics as nag. What can be accomplished with The quantum systems that can’t be restricts what can be done. done in a classical world? Explore what can be done with quantum systems, instead of being satisfied with what Nature hands us. Quantum engineering Metrology Taking the measure of things The heart of physics New physics Quantum mechanics as Old physics liberator. Quantum Explore what can be mechanics as nag. done with quantum The uncertainty systems, instead of principle being satisfied with restricts what can what Nature hands us. be done. Quantum engineering

Old conflict in new guise Measuring a classical parameter Phase shift in an (optical) interferometer Readout of anything that changes optical path lengths Michelson-Morley experiment Gravitational-wave detection Planck-scale, holographic uncertainties in positions

Torque on or free precession of a collection of spins Magnetometer Atomic clock Lectures 1 and 2

Force on a linear system Gravitational-wave detection Accelerometer Gravity gradiometer Electrometer Lecture 3 Strain meter II. Squeezed states and optical interferometry

Oljeto Wash Southern Utah (Absurdly) high-precision interferometry

Hanford, Washington

The LIGO Collaboration, Rep. Prog. Phys. 72, 076901 (2009).

Laser Interferometer Gravitational Observatory (LIGO)

4 km

Livingston, Louisiana (Absurdly) high-precision interferometry Initial LIGO Hanford, Washington

Laser Interferometer Gravitational Observatory (LIGO)

High-power, Fabry- Perot-cavity 4 km (multipass), power- recycled interferometers Livingston, Louisiana (Absurdly) high-precision interferometry Advanced LIGO Hanford, Washington

Currently a factor of 3 short of this design goal.

Laser Interferometer Gravitational Observatory (LIGO)

High-power, Fabry- Perot-cavity 4 km (multipass), power- and signal-recycled, squeezed-light interferometers Livingston, Louisiana Mach-Zender interferometer

C. M. Caves, PRD 23, 1693 (1981). Squeezed states of light Squeezed states of light

Groups at Australian National University, Hannover, and Squeezing by a factor of about 3.5 Tokyo have achieved up to 15 dB of squeezing at audio G. Breitenbach, S. Schiller, and J. Mlynek, frequencies for use in Advanced LIGO, VIRGO, and GEO. Nature 387, 471 (1997). Fabry-Perot Michelson interferometer

Motion of the mirrors produced by a induces a transition from the symmetric mode to the antisymmetric mode; the resulting tiny signal at the vacuum port is contaminated by that entered the vacuum port. Squeezed states and optical interferometry

K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf. R. Adhikari, K. McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nature Physics 4, 472 (2008).

44% improvement in displacement sensitivity Squeezed states for optical interferometry

H. Vahlbruch, A. Khalaidovski, N. Lastzka, C. Graef, K. Danzmann, and R. Schnabel, Classical and 27, 084027 (2010).

9dB below shot noise from 10 Hz to 10 kHz Squeezed states and optical interferometry

GEO 600 laser interferometer

The LIGO Scientific Collaboration, Nature Physics 7, 962 (2011).

Up to 3.5dB improvement in sensitivity in the shot-noise- limited frequency band Squeezed states and optical interferometry

~ 2 dB of shot-noise reduction

Squeezed light in the LIGO Hanford detector

The LIGO Scientific Collaboration, Nat. Phot. 7, 613 (2013). Quantum limits on optical interferometry

Quantum Noise Limit (Shot-Noise Limit)

As much power in the squeezed Heisenberg Limit light as in the main beam III. Ramsey interferometry, cat states, and spin squeezing

Truchas from East Pecos Baldy Sangre de Cristo Range Northern New Mexico Ramsey interferometry

N independent “atoms”

Frequency measurement Time measurement Clock synchronization Cat-state Ramsey interferometry J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, Phys. Rev. A 54, R4649 (1996).

Fringe pattern with period 2π/N

N cat-state atoms Optical interferometry Ramsey interferometry

Quantum Noise Limit (Shot-Noise Limit)

Heisenberg Limit

Something’s going on here. Squeezed-state optical Cat-state Ramsey interferometry interferometry Entanglement before “beamsplitter”

Between arms Between atoms (wave or modal entanglement) (particle entanglement)

Between photons Between arms (particle entanglement) (modal entanglement) Squeezed-state optical Cat-state Ramsey interferometry interferometry Entanglement after “beamsplitter”

Between arms Between atoms (wave or modal entanglement) (particle entanglement)

Between photons Between arms (particle entanglement) (modal entanglement) Spin-squeezing Ramsey interferometry J. Ma, X. Wang, C. P. Sun, and F. Nori, Phys. Rep. 509, 89‒165 (2011).

Heisenberg Limit

This is really a cat state. Spin-squeezing Ramsey interferometry

What’s squeezed? The +y spin state has N particles; the –y spin state has single-mode squeezing. This is like the state of the two arms prior to the beamsplitter in an optical interferometer. The up and down spin states have correlated squeezing like that in the arms of a squeezed-state optical interferometer.

What’s entangled? No entanglement of +y and –y spin states. Modal entanglement of up and down spin states. Particle entanglement. Squeezed-state optical Spin-squeezing Ramsey interferometry interferometry

Entanglement

Between arms Between atoms (wave or modal entanglement) (particle entanglement)

Between photons Between arms (particle entanglement) (modal entanglement) Role of entanglement

Entanglement is a resource … for getting my paper into Nature.

Don’t accept facile explanations. Ask questions. Transition

Telling stories is what physics is about.

Lecture 1 has been about understanding fringe patterns and sources of noise and designing devices to improve phase sensitivity based on this understanding. This is telling stories.

Lecture 2 is about proving that the stories aren’t fooling us.

Which is better, stories or proofs? You need them both, but stories are going to get you farther. Quantum metrology: An information-theoretic perspective

Lecture 2 I. Quantum Cramér-Rao Bound (QCRB) II. Making quantum limits relevant. Loss and decoherence III. Beyond the Heisenberg limit. Nonlinear interferometry

Carlton M. Caves Center for Quantum Information and Control, University of New Mexico

http://info.phys.unm.edu/~caves

Center for Quantum Information and Control Transition

Telling stories is what physics is about.

Lecture 1 has been about understanding fringe patterns and sources of noise and designing devices to improve phase sensitivity based on this understanding. This is telling stories.

Lecture 2 is about proving that the stories aren’t fooling us.

Which is better, stories or proofs? You need them both, but stories are going to get you farther. I. Quantum Cramér-Rao Bound (QCRB)

Cable Beach Western Australia Quantum information version of interferometry

Quantum noise limit

Quantum circuits cat state N = 3

Heisenberg limit

Fringe pattern with period 2π/N Cat-state interferometer

State Measurement preparation

Single- parameter estimation S. L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. Phys. 247, 135 (1996). Heisenberg V. Giovannetti, S. Lloyd, and L. Maccone, PRL 96, 041401 (2006). limit S. Boixo, S. T. Flammia, C. M. Caves, and JM Geremia, PRL 98, 090401 (2007).

Separable inputs

Generalized uncertainty principle Quantum Cramér-Rao bound Achieving the Heisenberg limit

cat Proof of state QCRB

Is it entanglement? It’s the entanglement, stupid.

But what about?

For metrology, entanglement is part of the story, but only part. We need a generalized notion of entanglement/resources that includes information about the physical situation, particularly the relevant Hamiltonian. II. Making quantum limits relevant. Loss and decoherence

Bungle Bungle Range Western Australia Making quantum limits relevant

Quantum circuits

The serial resource, T, and The serial resource, T, and the parallel resource, N, are the parallel resource, N, are equivalent and not equivalent and not interchangeable, interchangeable, physically. mathematically.

Information science Physics perspective perspective Distinctions between different Platform independence physical systems Making quantum limits relevant. One metrology story

A. Shaji and C. M. Caves, PRA 76, 032111 (2007). Making quantum limits relevant Rule of thumb for photon losses for large N S. Knysh, V. N. Smelyanskiy and G. A. Durkin, PRA 83, 021804(R) (2011).

Heisenberg limit: less than one photon lost.

Typically, beat shot noise by square of root of fractional loss. Quantum limit on practical optical interferometry 1. Cheap photons from a laser (coherent state) 2. Low, but nonzero losses on the detection timescale 3. Beamsplitter to make differential phase detection insensitive to laser fluctuations

Freedom: state input to the second input port; optimize relative to a mean-number constraint. Entanglement: mixing this state with coherent state at the beamsplitter.

Generalized uncertainty principle QCRB

Optimum achieved by M. D. Lang and C. M. Caves, differenced photodetection in a PRL 111, 17360 (2013). Mach-Zehnder configuration. Achieved by squeezed vacuum into the second input port Practical optical interferometry: Photon losses M. D. Lang , UNM PhD dissertation, 2015.

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, Nat. Phys. 7, 406‒411 (2011).

Z. Jiang, PRA 89, 032128 (2014).

Optimum achieved by differenced photodetection in a Mach-Zehnder configuration. III. Beyond the Heisenberg limit. Nonlinear interferometry

Echidna Gorge Bungle Bungle Range Western Australia Beyond the Heisenberg limit

The purpose of theorems in physics is to lay out the assumptions clearly so one can discover which assumptions have to be violated. S. Boixo, S. T. Flammia, C. M. Caves, and Improving the scaling with N JM Geremia, PRL 98, 090401 (2007).

Cat state does the job. Nonlinear Ramsey interferometry

Metrologically relevant k-body coupling Improving the scaling with N S. Boixo, A. Datta, S. T. Flammia, A. without entanglement Shaji, E. Bagan, and C. M. Caves, PRA 77, 012317 (2008).

Product Product input measurement Improving the scaling with N without entanglement. Two-body couplings

S. Boixo, A. Datta, S. T. Flammia, A. Shaji, E. Bagan, and C. M. Caves, PRA 77, 012317 (2008); M. J. Woolley, G. J. Milburn, and C. M. Caves, NJP 10, 125018 (2008). Loss and decoherence? Improving the scaling with N without entanglement. Two-body couplings

Super-Heisenberg scaling from nonlinear dynamics (N- enhanced rotation of a spin coherent state), without any particle entanglement

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, PRL 101, 040403 (2008); A. B. Tacla, S. Boixo, A. Datta, A. Loss and decoherence? Shaji, and C. M. Caves, PRA 82, 053636 (2010). Improving the scaling with N without entanglement. Optical experiment

M. Napolitano, M. Koschorreck, B. Dubost, N. Behbood, R. J. Sewell, and M. W. Mitchell, Nature 471, 486 (2011). Quantum metrology: An information-theoretic perspective

Lecture 3 I. Introduction. What’s the problem? II. Standard quantum limit (SQL) for force detection. The right wrong story III. Beating the SQL. Three strategies

Carlton M. Caves Center for Quantum Information and Control, University of New Mexico Centre for Engineered Quantum Systems, University of Queensland http://info.phys.unm.edu/~caves

Center for Quantum Information and Control I. Introduction. What’s the problem?

Pecos Wilderness Sangre de Cristo Range Northern New Mexico Measuring a classical parameter Phase shift in an (optical) interferometer Readout of anything that changes optical path lengths Michelson-Morley experiment Gravitational-wave detection Planck-scale, holographic uncertainties in positions

Torque on or free precession of a collection of spins Magnetometer Atomic clock Lectures 1 and 2

Force on a linear system Gravitational-wave detection Accelerometer Gravity gradiometer Electrometer Lecture 3 Strain meter (Absurdly) high-precision interferometry for force sensing Hanford, Washington

The LIGO Collaboration, Rep. Prog. Phys. 72, 076901 (2009).

Laser Interferometer Gravitational Observatory (LIGO)

4 km

Livingston, Louisiana (Absurdly) high-precision interferometry for force sensing Initial LIGO Hanford, Washington

Laser Interferometer Gravitational Observatory (LIGO)

High-power, Fabry- Perot-cavity 4 km (multipass), power- recycled interferometers Livingston, Louisiana (Absurdly) high-precision interferometry for force sensing Advanced LIGO Hanford, Washington

Laser Interferometer Gravitational Observatory (LIGO)

High-power, Fabry- Perot-cavity 4 km (multipass), power- and signal-recycled, squeezed-light interferometers Livingston, Louisiana Opto,atomic,electro micromechanics

30 μm long 170 nm wide 140 nm thick

Beam microresonator

10 μm T. Rocheleau, T. Ndukum, C. Macklin , J. B. Hertzberg, A. A. Clerk, and K. C. Schwab, Nature 463, 72 (2010).

Atomic force microscope

Dielectric micromembrane

J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nature Physics 6, 707 (2010). Opto,atomic, electro micromechanics

Zipper-cavity microresonator

Drum microresonator

A. D. O’Connell et al., Nature 464, 697 (2010).

M. Eichenfield, R. Camacho, J. Chan, K. J. Vahala, and O. Painter, Nature 459, 550 (2009).

A. Schliesser and T. J. Kippenberg, Advances in Atomic, Molecular, and Optical Physics, Vol. 58, (Academic Press, San Diego, 2010), p. 207. Toroidal microresonator Mechanics for force sensing

T. J. Kippenberg and K. J. Vahala, Science 321, 172 (2008). Standard quantum limit (SQL) Wideband detection of force f on free mass m LIGO interferometer

Back action Standard quantum limit (SQL) Narrowband, on-resonance detection of force f on oscillator of mass m and resonant frequency ω0 Nanoresonator

Back action? SQL Wideband force f on free mass m

On-resonance force f on oscillator of mass m and resonant frequency ω0

It’s wrong. It’s not even the right wrong story.

The right wrong story. Waveform estimation. II. Standard quantum limit (SQL) for force detection. The right wrong story

San Juan River canyons Southern Utah SQL for force detection

Monitor position

Back-action force Langevin force

measurement (shot) noise Interferometric — readout

Laser Interferometric — readout

Laser Interferometric — readout

Vacuum input port

measurement Laser (shot) noise

Back-action noise

If shot noise dominates, squeeze the phase quadrature. SQL for force detection

Back-action force Time domain Langevin force

measurement noise

Frequency domain

Back-action force

Langevin force measurement noise Noise-power spectral densities

Zero-mean, time-stationary random process u(t)

Noise-power spectral density of u SQL for force detection

Back-action force

Langevin force measurement noise SQL for force detection Langevin force SQL for force detection

The right wrong story.

In an opto-mechanical setting, achieving the SQL at a particular frequency requires squeezing at that frequency, and achieving the SQL over a wide bandwidth requires frequency-dependent squeezing. III. Beating the SQL. Three strategies

Truchas from East Pecos Baldy Sangre de Cristo Range Northern New Mexico Beating the SQL. Strategy 1

1. Couple parameter to h, and monitor observable o conjugate to h. 2. Arrange that h and o are conserved in the absence of the parameter interaction; o is the simplest sort of quantum nondemolition (QND) or back-action-evading (BAE) observable. 3. Give o as small an uncertainty as possible, thereby giving h as big an uncertainty as possible (back action).

Strategy 1. Monitor a quadrature component.

Downsides 1. Detect only one quadrature of the force. 2. Mainly narrowband (no convenient free-mass version). 3. Need new kind of coupling to monitor oscillator. Strategy 2. — Interferometric readout

Vacuum input port

Output noise Laser

All the output noise comes from the (frequency-dependent) purple quadrature. Squeeze it. W. G. Unruh, in , Experimental Gravitation, and Measurement Theory, edited by P. Meystre and M. O. Scully (Plenum, 1983), p. 647; F. Ya. Khalili, PRD 81, 122002 (2010). Beating the SQL. Strategy 2

Strategy 2. Squeeze the entire output noise by correlating the measurement and back-action noise. Quantum Cramér-Rao Bound (QCRB)

Single-parameter estimation: Bound on the error in estimating a classical parameter that is coupled to a quantum system in terms of the inverse of the quantum Fisher information.

Multi-parameter estimation: Bound on the covariance matrix in estimating a set of classical parameters that are coupled to a quantum system in terms of the inverse of a quantum Fisher-information matrix.

Waveform estimation: Bound on the continuous covariance matrix for estimating a continuous waveform that is coupled to a quantum system in terms of the inverse of a continuous, two-time quantum Fisher-information matrix. M. Tsang, H. M. Wiseman, and C. M. Caves, Waveform QCRB. PRL 106, 090401 (2011). Spectral uncertainty principle

Prior-information term

At frequencies where there is little prior information,

Minimum-uncertainty noise

No hint of SQL—no back-action noise, only measurement noise—but can the bound be achieved? Beating the SQL. Strategy 3 Strategy 3. Quantum noise cancellation (QNC) using oscillator and negative-mass oscillator.

Primary oscillator Negative-mass oscillator

Monitor collective position Q

Conjugate pairs QCRB

Oscillator pairs M. Tsang and C. M. Caves, Quantum noise cancellation PRL 105,123601 (2010).

Conjugate pairs

Oscillator pairs

Paired sidebands about a carrier frequency Paired collective spins, polarized along opposite directions

W. Wasilewski , K. Jensen, H. Krauter, J. J. Renema, M. V. Balbas, and E. S. Polzik, PRL 104, 133601 (2010). That’s all. Thanks for your attention.

Tent Rocks Kasha-Katuwe National Monument Northern New Mexico Using diagrams

Cat-state interferometer

2.I

Cat-state interferometer 2.II

C. M. Caves and A. Shaji, Opt. Commun. 283, 695 (2010) . Proof of QCRB. Setting Proof of QCRB. Classical CRB Proof of QCRB. Classical CRB Proof of QCRB. Classical Fisher information Proof of QCRB. Quantum mechanics Proof of QCRB. Quantum mechanics Proof of QCRB. Quantum mechanics